1. (10 points) Consider a continuous time Markov chain model for the weather in LA with...

Transcribed Image Text:

1. (10 points) Consider a continuous time Markov chain model for the weather in LA with three states: 1 = sunny, 2 = cloudy, 3 = rainy. The weather stays sunny for an exponentially distributed number of days with mean 3, then becomes cloudy. It stays cloudy for an expo- nentially distributed number of days with mean 4, then rain comes. The rain lasts for an exponentially distributed number of days with mean 1, then sunshine returns. (a) Find the stationary distribution of this Markov chain. (b) Consider the discrete-time jump process associated with this chain. Find the transition probability matrix of the jump process. (c) Given that it is sunny right now, find the probability that it will not rain for 3 days. 1. (10 points) Consider a continuous time Markov chain model for the weather in LA with three states: 1 = sunny, 2 = cloudy, 3 = rainy. The weather stays sunny for an exponentially distributed number of days with mean 3, then becomes cloudy. It stays cloudy for an expo- nentially distributed number of days with mean 4, then rain comes. The rain lasts for an exponentially distributed number of days with mean 1, then sunshine returns. (a) Find the stationary distribution of this Markov chain. (b) Consider the discrete-time jump process associated with this chain. Find the transition probability matrix of the jump process. (c) Given that it is sunny right now, find the probability that it will not rain for 3 days.